v2010.10.26 - Convex Optimization

100 CHAPTER 2. CONVEX GEOMETRY2.7.1 Cone definedA set X is called, simply, cone if and only ifΓ ∈ X ⇒ ζΓ ∈ X for all ζ ≥ 0 (174)where X denotes closure of cone X . An example of such a cone is the unionof two opposing quadrants; e.g., X ={x∈ R 2 | x 1 x 2 ≥0} which is not convex.[371,2.5] Similar examples are shown in Figure 33 and Figure 37.All cones can be defined by an aggregate of rays emanating exclusivelyfrom the origin (but not all cones are convex). Hence all closed cones containthe origin 0 and are unbounded, excepting the simplest cone {0}. Theempty set ∅ is not a cone, but its conic hull is;cone ∅ = {0} (104)2.7.2 Convex coneWe call set K a convex cone iffΓ 1 , Γ 2 ∈ K ⇒ ζΓ 1 + ξΓ 2 ∈ K for all ζ,ξ ≥ 0 (175)id est, if and only if any conic combination of elements from K belongs to itsclosure. Apparent from this definition, ζΓ 1 ∈ K and ξΓ 2 ∈ K ∀ζ,ξ ≥ 0 ;meaning, K is a cone. Set K is convex since, for any particular ζ,ξ ≥ 0µζΓ 1 + (1 − µ)ξΓ 2 ∈ K ∀µ ∈ [0, 1] (176)because µζ,(1 − µ)ξ ≥ 0. Obviously,{X } ⊃ {K} (177)the set of all convex cones is a proper subset of all cones. The set ofconvex cones is a narrower but more familiar class of cone, any memberof which can be equivalently described as the intersection of a possibly(but not necessarily) infinite number of hyperplanes (through the origin)and halfspaces whose bounding hyperplanes pass through the origin; ahalfspace-description (2.4). Convex cones need not be full-dimensional.